# Rotation and translation of a cylinder

Hannisch

## Homework Statement

A homogenous cylinder with radius R and mass M is resting, with the axis vertical, on a horizontal surface, which can be asumed to be completely slippery. A string is partially wound around the cylinder. The string then runs over a pulley at the end of the table, in the same height as the winding, and then on to a weight with mass m hanging in its other end. Determine the angular acceleration and the tension in the string when the system is moving. The moment of intertia of the pulley can be neglected.

F = ma

RF = Iα

## The Attempt at a Solution

I'm going crazy with this, because it's not really that hard of a problem..

Okay, if I draw the force diagrams, I have the tension T on the cylinder, which is the force that will create its angular acceleration and I have T up and mg down on the mass m. The tension should also be the force that's accelerating the centre of mass of M. I'd say the translational acceleration of M needs to be the same as that for m.

m: mg - T = ma
M: T = Ma
rotation: RT = Iα

I for a cyldinder = MR2/2

from rotation: T = MRα/2

from M: a = T/M

a = Rα/2

Putting these in the m-equation: mg - MRα/2 = mRα/2

mg = Rα(M + m)/2

2mg/R(M + m) = α

And this is incorrect. There's something I'm missing and I can't see it - it is driving me insane. The correct answer is supposed to be α = 2mg/R(M + 3m). So close, yet sooo far away.

Could anyone tell me what I'm not seeing, please?